Chapter 1Kriging, Splines, ConditionalSimulation, Bayesian Inversionand Ensemble Kalman Filtering

Olivier Dubrule

Abstract This chapter discusses, from a theoretical point of view, how the geo-statistical approach relates to other commonly-used models for inversion or dataassimilation in the petroleum industry. The formal relationship between pointKriging and splines or radial basis functions is first presented. The generalizationsof Kriging to the estimation of average values or values affected by measurementerrors are also addressed. Two algorithms are often used for conditional simulation:the “rough plus smooth” approach consists of adding a smooth correction to anon-conditional simulation, whilst sequential Gaussian simulation allows thepoint-by-point construction of the realizations. As with Kriging, conditional sim-ulation can be applied to average values or to data affected by measurement errors.Geostatistical inversion generates high-resolution realizations of vertical impedancetraces constrained by seismic amplitudes. If the relationship between impedanceand amplitude data is linearized, geostatistical inversion is a particular case ofBayesian inversion. Because of the non-linearity of production data vis-à-vis thevariables of the earth model, their assimilation is harder than that of seismic data.Ensemble Kalman filtering, if considered from a geostatistical viewpoint, consistsof using a large number—or ensemble—of realizations to calculate empiricalcovariances between the dynamic data and the parameters of the geostatisticalmodel. These covariances are then used in the equations for interpolating themismatch between simulated and new production data using a coKriging-likeformalism. Interestingly, most of these techniques can be expressed using the samegeneric equation by which an initial model not honouring some newly arrived datais made conditional to these data by adding a (co-)Kriged interpolation of the datamismatches to the initial model. In spite of their similar equations, Bayesianinversion, geostatistics and ensemble Kalman filtering have a different approach tothe inference of the covariance models used by these equations.

Keywords Dual Kriging ⋅ Radial basis functions ⋅ Geostatistical inversionEnergy-based methods ⋅ Prediction error filter

O. Dubrule (✉)Imperial College London, London, UKe-mail: o.dubrule@imperial.ac.uk

© The Author(s) 2018B. S. Daya Sagar et al. (eds.), Handbook of Mathematical Geosciences,https://doi.org/10.1007/978-3-319-78999-6_1

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1.1 Introduction

Fifty years ago, when geostatistics was pioneered by Matheron (1971), its mainapplications were Kriging and the change of support for mining applications. At thetime, geostatistics was presented as a new discipline, without much reference to itsrelationships with other mathematical interpolation and modeling techniques. Thishas now changed as the relationships between geostatistics and such techniques assplines, regularization, Bayesian inversion, or ensemble Kalman filtering havebecome clearer. This convergence is fascinating and has led to many significantdevelopments allowing the integration of multi-disciplinary data into 3-D geosta-tistical earth models.

This chapter discusses approaches for generating 2-D or 3-D subsurface modelsconstrained by geological (wells), seismic or dynamic data. In spite of the wealth ofdata available, the uncertainty on the 3-D earth model remains high in most cases.Approaches that are designed to generate one unique “deterministic” model oftenpick the smoothest one. This is not realistic in situations where the Earth Model isused for flow simulation, as the results are biased if the model heterogeneities arenot representative of that of the actual reservoir. More generally, non-linear oper-ations, such as the application of cut-offs, may give biased results when applied todeterministic smooth models such as those produced by Kriging.

The multi-realization approach is now routinely applied to subsurface parametersinversion. Looking at the mean provides much less information than looking at amovie of realizations. …By construction, each of the realizations captures theessential random fluctuations of the actual field from which the data were extracted(Tarantola 2005). This is a fundamental change. The traditional inversion approachcould be formulated as “How to find an estimate of the spatial parameters which isas close as possible to the first guess values of these parameters and which provides,through forward modeling, an output which is as close as possible to the availabledata” (modified from Evensen 2007). These first guess values are usually a smooth(Kriging-like) spatial model of these parameters. Now the question has changed to“Find the probability density function (pdf) of 3-D models constrained by all theexisting data, and provide techniques for sampling realizations from this pdf”.

This chapter, written from a geostatistical perspective, discusses the convergencebetween the existing techniques.

Deterministic approaches such as Kriging, splines, regularization- orenergy-based methods generate a single model of the subsurface, which usuallyminimizes or maximizes an optimisation criterion. These approaches are closelyrelated and their formal relationships are discussed.

Geostatistical simulation is then revisited, and two key simulation algorithms arediscussed; The first one is sequential Gaussian simulation and the second one is the“rough plus smooth” combination of an unconditional simulation plus a smoothcorrection term. These two algorithms have helped bridge the gap between geo-statistics and inversion.

4 O. Dubrule

Two successful approaches are then discussed for integrating seismic anddynamic data into the earth model. Rather than using an approach merely based onstatistical correlations between data and model parameters, it is assumed that thereexists a deterministic relationship (or forward model) between model parametersand data, possibly including a random error.

The first approach, geostatistical inversion, produces reservoir-scale models ofacoustic or elastic parameters constrained by single- or multi-offset seismicamplitude data. The value of using sequential Gaussian simulation to calculateseismically-constrained realizations is discussed. In situations where the forwardmodel is linear, geostatistical inversion can be formulated as a particular case ofBayesian seismic inversion.

The second approach, ensemble Kalman filtering, consists of sequentiallyupdating an “ensemble” of geostatistical realizations using dynamic data as they areacquired in time. The key idea here is to statistically derive the covariance terms ofthe equation used in Bayesian inversion from an ensemble of realizations ratherthan from a theoretical covariance model. The formal relationship betweenensemble Kalman filtering and co-Kriging is discussed.

Most of the above techniques can be shown to use the same kind of formalism,where the mismatch between newly arrived data and the current model is inter-polated and used to update this model.

One of the conclusions of this chapter is that the equations of Bayes, geostatisticsor ensemble Kalman filtering are closely related. However, this relationship ismostly formal as the three techniques differ in their approach to the covariancesused in the equations. Geostatisticians first fit a model to the data, whilst Bayesiansstart from a model based on general “prior” information. Only later in the processdo they introduce the well data. And ensemble Kalman filtering directly uses theexperimental covariances calculated from the realizations of the ensemble.

The topic of joint inversion of seismic and dynamic data is not discussed here, inspite of the interesting on-going developments in 4-D seismic data inversion. Thisis because the objective of this chapter is to address formal relationships betweenthe different formalisms rather than discuss specific applications.

1.2 Deterministic Aspects of Geostatistics

1.2.1 Simple Stationary Kriging

The basic model used by geostatistics is that of stationary random functions of order2: a spatial property z xð Þ at location x is represented by a random function Z xð Þ,which is assumed to follow a trend m xð Þ and a stationary covariance C hð Þ

1 Kriging, Splines, Conditional Simulation, Bayesian Inversion … 5

mðxÞ=E Z xð Þð Þ ð1:1aÞ

C hð Þ=E Z xð ÞZ x+hð Þð Þ−E Z xð Þð ÞE Z x+hð Þð Þ ð1:1bÞ

At each unsampled location x, the value of Z xð Þ is estimated by a linear com-bination Zk xð Þ of the values Zi =Z xið Þ at the n data points xið Þi=1, ..., n. Kriging is thebest linear unbiased estimator, in the sense that it is unbiased and that it minimizesthe estimation variance. If the trend m xð Þ is known at each location x, the simpleKriging (Chilès and Delfiner 2012, p. 151) system of equations is obtained

Zk xð Þ−m xð Þ= ∑n

i=1λi Zi −m xið Þð Þ ð1:2aÞ

with ∑n

i=1λiC xi − xj� �

=C x− xj� �

for j∈ 1, . . . , nð Þ ð1:2bÞ

1.2.2 Kriging with Intrinsic Random Functions of Order k

Matheron (1973) generalized the above model to that of Intrinsic Random Func-tions of Order k (IRF-k), where the definition of the variogram as a generalizedcovariance of order zero and of generalized covariances of order k leads to a modelbased on the stationarity of generalized increments of order k.

With k-IRFs, the model only considers linear combinations of Z xð Þ that filterpolynomials of order k (such polynomials being likely to represent a trend). SimpleKriging is not applicable any more. For instance, if k = 1 in two dimensions, and ifK hð Þ designates the generalized covariance of order k (GC-k), the kriging systembecomes

ZkðxÞ= ∑n

i=1λiZi ð1:3aÞ

with ∑n

i=1λiK xi − xj� �

+ μ0 + μ1xj1 + μ2xj2 =K x− xj� �

for j∈ 1, . . . , nð Þ

and ∑n

i=1λi =1 ∑

n

i=1λixi1 = x1 ∑

n

i=1λixi2 = x2

ð1:3bÞ

where the coordinates of each point x of the plane are written as x= x1, x2ð Þ.

6 O. Dubrule

1.2.3 Kriging Extensions

The goal here is not to discuss the details of Kriging, as there are plenty of excellenttextbooks for this (Chilès and Delfiner 2012, p. 150). However, two features ofKriging deserve to be discussed, as they facilitate the understanding of the rela-tionship between Kriging, splines and Bayesian approaches.

1.2.3.1 Generalization of Kriging to the Interpolationof Average Values

Kriging is a linear interpolator. The data used by Kriging do not have to be pointvalues, but they can be any linear function of the parameters of interest; Hansenet al. (2006) call these “volume support data”. In particular, Kriging can be used toestimate the average value of a parameter ZðvxÞ at a location x by a linear com-bination volume of support data ZðvxiÞ (Chilès and Delfiner 2012, p. 198)

Zk vxð Þ= ∑n

i=1λiZðvxiÞ ð1:4Þ

This property of Kriging, extensively used in mining applications, is of signif-icant interest in the context of linear inversion of volume support data (Hansen et al.2006). The Kriging equations associated with Eq. 1.4 are not given here, as they area bit heavy, but conceptually simple thanks to the linear property of Kriging.

1.2.3.2 Error CoKriging

Error coKriging (Dubrule 2003) is a generalization of Kriging to the situation wheremeasurements Yi of the parameter Zi at data points xi are affected by an unbiasedrandom error

Yi = Zi + εi withE εið Þ=0 andVar εið Þ=Cεi ð1:5Þ

In this situation, error coKriging allows the estimation of Z xð Þ at any unsampledlocation x from a linear combination of values Yi (the random measurement errorattached to each data can be zero or not) (Dubrule 2003; Hansen et al. 2006; Chilèsand Delfiner 2012, p. 216)

Zk xð Þ= ∑n

i=1λiYi ð1:6Þ

1 Kriging, Splines, Conditional Simulation, Bayesian Inversion … 7

1.2.3.3 Dual Kriging

If a global neighborhood is used, that is if all the available data are used to estimateZ xð Þ at every single location x, the Kriging equations (Eq. 1.3) can be inverted toobtain the dual Kriging system (for interpolation in the case of Kriging andsmoothing in the case of error coKriging). For example, in two dimensions for ak-IRF of order 1

zk xð Þ= zk x1, x2ð Þ= a0 + a1x1 + a2x2 + ∑n

i=1biK x− xið Þ ð1:7Þ

where the conditions on the coefficients ða0, a1, a2, b1, . . . , bnÞ are different forKriging and error coKriging (Dubrule 1983)

Kriging: ∑n

i=1bi = ∑

n

i=1bixi1 = ∑

n

i=1bixi2 = 0 and zk xi1, xi2ð Þ= zi ð1:8Þ

Error coKriging: ∑n

i=1bi = ∑

n

i=1bixi1 = ∑

n

i=1bixi2 = 0 and zk xi1, xi2ð Þ+ biCεi = yi

ð1:9Þ

1.2.4 Kriging and Splines

1.2.4.1 Interpolating Splines

Splines are a popular method for deterministic interpolation and approximation(Micula and Micula 1999). In 2-D, interpolating splines calculate a functionhonouring the data and minimizing an energy functional. Harmonic splines mini-mize the stretching energy of a membrane while biharmonic splines minimize thebending energy of an elastic plate. The biharmonic spline function can be writtenusing a similar expression as Eq. 1.7 (Duchon 1975), but with a specific model forthe generalized covariance function

K x− xið Þ= x1 − xi1ð Þ2 + x2 − xi2ð Þ2� �

Logffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix1 − xi1ð Þ2 + x2 − xi2ð Þ2

q� �ð1:10Þ

Splines and Kriging are a particular case of a more general class of interpolators,called radial basis functions (Billings et al. 2002a, b). With splines, the polynomialin Eq. 1.7 belongs to the kernel of the operator T that is minimized by the splinefunction (T is the gradient for harmonic splines and the laplacian for biharmonic

8 O. Dubrule

splines), whilst the function K hð Þ is the Green function associated with the operatorT 0T , where T 0 is the transposed operator of T (Matheron 1981a)

T 0TK hð Þ= δ ð1:11Þ

where δ is the Dirac Function. Choosing the energy functional minimized bysplines is equivalent to fixing the degree of the trend function and the generalizedcovariance model for Kriging. For harmonic splines, these are respectively a con-stant and the De Wijs variogram in Logh (Chilès and Delfiner 2012, p. 94).

The consequence of Eq. 1.11 on the spectral density of the generalizedcovariance K hð Þ is straightforward. For example, the spectral densities associatedwith the harmonic and biharmonic splines are power laws, representing fractalmodels. Szeliski and Terzopoulos (1989) and Micula and Micula (1999) discussthis relationship between Splines and fractals.

1.2.4.2 Smoothing Splines

Smoothing splines are used in situations where measurements at data points areaffected by a random error (Eq. 1.5). In two dimensions, they compute a functionf x1, x2ð Þ minimizing the sum of a spline energy functional plus a weighted distanceto the n data

Tfk k2 + θ ∑n

i=1

f xi1, xi2ð Þ− yið Þ2Cεi

ð1:12Þ

The smoothing biharmonic spline function has the same expression as that ofKriging and error coKriging (Eq. 1.7) but with the following relationships

∑n

i=1bi = ∑

n

i=1bixi1 = ∑

n

i=1bixi2 = 0 and f x1i, xi2ð Þ+ bi

Cεi

θ= yi ð1:13Þ

Smoothing biharmonic splines are identical to error Cokriging as long as thegeneralized covariance used by error Cokriging is the function θK x− xið Þ, whereK x− xið Þ is given by Eq. 1.10 (Matheron 1981a; Dubrule 2003). This is a generalrelationship between smoothing splines and coKriging, which are formally equiv-alent if the generalized covariance K hð Þ is that satisfying Eq. 1.11, with thecoefficient of K hð Þ equal to the smoothing parameter θ of Eq. 1.12.

1 Kriging, Splines, Conditional Simulation, Bayesian Inversion … 9

1.2.4.3 Kriging and Regularization—The Discrete Case

The discrete case is the situation where interpolation is performed at the nodes of aregular grid and each data point is located at one of the nodes of this grid. If p is thetotal number of grid nodes, the number n of data points is such that n< p.

In the discrete case, Matheron (1981b) also demonstrated the equivalencebetween splines and Kriging, and between smoothing splines and error coKriging.Both the Kriged and spline values zu minimize

∑p

u, v=1zuBuvzv + ∑

n

i=1

zi − yið Þ2Cεi

ð1:14Þ

where the u and v indices designate all the p grid points where the interpolationtakes place, whilst i indices designate the n data points. The minimization ofEq. 1.14 is performed according to the unknown values zu at all grid nodes (in-cluding those unknown values zi where a data point with measured value yi ispresent). The first term of Eq. 1.14 can be interpreted as a quadratic energy functiontraditionally used in inverse problems. In the regularization context, the choice ofthis quadratic form is driven by smoothing considerations, often using Briggs’ finitedifference Laplacian (or spline) “roughening” operator (Briggs 1974; Bolondi et al.1976). Seen from the geostatistical perspective, Buv is the inverse of the covariancematrix in the stationary case and a pseudo-inverse of the generalized covariancematrix in the k-IRF case (Matheron 1981b). Equation 1.14 confirms the clearrelationship between the inverse of the (generalized) covariance and the splinedifferential operator.

Kriging can thus be formalized in the frame of energy-based estimation tech-niques such as splines. This comes from the relationship between the inverse of thecovariance function and the roughening filter implicit in the quadratic regularizationterm. It will be shown below that the regularization term can also be regarded, in theBayesian inversion context, as an expression of the prior knowledge about thevariable under study.

1.2.5 Kriging and Bayesian Inversion

1.2.5.1 Bayesian Linear Inversion

Here it may be useful to recall the general expression of the posterior mean andcovariance in the case of Bayesian linear inversion of a multigaussian function.A very good reference for this is Tarantola (2005).

In the discrete case, consider a stationary multigaussian random vector z ofdimension p containing the grid values zu over a two or three-dimensional regulargrid of size p. Assume also that a vector y contains the n data yi. It is assumed again

10 O. Dubrule

that the data are affected by an error vector ε of dimension n, and also that thesedata are a linear function of the p values of z over the grid

y=Fz+ ε ð1:15Þ

where the vector ε has mean zero and covariance matrix Cε and F is a matrix ofdimension n× p. In the multigaussian case, thanks to the Bayes formula relating theposterior pdf fpost zð Þ to the prior pdf fprio zð Þ and the likelihood function g y zjð Þ, theprior mean vector m (dimension p) and covariance matrix C (dimension p× p) of zare updated using the information brought by the data vector y

fpost zð Þ∝fprio zð Þg y ̸zð Þ∝ exp z−mð Þ′C − 1 z−mð Þh i

× exp y−Fzð Þ′C − 1ε y−Fzð Þ

h ið1:16Þ

fpost zð Þ is a multigaussian function with the mean vector

mpost =m+C F′ FCF′ +Cε

� �− 1y−F mð Þ ð1:17Þ

and the covariance matrix

Cpost =C−CF′ FCF′ +Cε

� �− 1FC ð1:18Þ

1.2.5.2 Kriging and Bayesian Inversion

Equation 1.17 can also be written

mpost =m+Λ y−Fmð Þ= ðI −ΛFÞm+Λy ð1:19Þ

with

Λ=C F′ FCF′ +Cε

� �− 1 ð1:20Þ

In can be checked that Λ is also the p× n matrix giving at each line u the nsimple Kriging (or error coKriging) weights associated with the Kriging of thevalue zu at node u. Comparing the first part of Eq. 1.19 with Eq. 1.2 shows that, inthe multigaussian case, mpost is equal to simple Kriging and that the matrix Cpost

contains the variances and covariances of simple Kriging at each node u of theregular grid.

1 Kriging, Splines, Conditional Simulation, Bayesian Inversion … 11

1.2.6 Energy-Based Versus Probabilistic Estimates

The minimization of Eq. 1.14 leads to either Kriging or splines if the (inverse of)the covariance (Kriging) and the differential operator (splines) are properly chosen.Minimizing the expression in Eq. 1.14 is equivalent to maximizing

exp − ∑p

u, v=1zuBuvzv + ∑

n

i=1

zi − yið Þ2Cεi

! !ð1:21Þ

This is also the expression (up to a multiplicative constant) of the conditionalmultivariate distribution in the multigaussian case, as given by Bayes theorem(Eq. 1.16), in the case where m=0, where the matrix Cε is diagonal and where thedata are point values. The first term represents the prior pdf and the second thelikelihood function. Kriging which is equal to the mean of the posterior pdf, alsomaximizes this pdf in the multigaussian case.

Expression (1.21) relates the world of energy functionals (such as splines) withthat of probability functions (such as Kriging). More generally regularization andmaximum a posteriori Bayesian estimates are identical if the prior covariance usedin Bayesian inversion is properly chosen. The equivalence between an energyfunction and a probability distribution is also used in statistical mechanics, as theprobability of a particular configuration is inversely related to its energy. Supposethat the vector z minimizes an energy functional E zð Þ. Using the results of Gemanand Geman (1984), Szeliski and Terzopoulos (1989) associate a probability to thisenergy through the Boltzmann (or Gibbs) distribution p zð Þ defined as

p zð Þ= 1Zexp −

E zð ÞT

� �ð1:22Þ

where Z and T are positive constants. If Bayes’ theorem is applied to the aboveprior pdf p zð Þ and the posterior pdf is maximized, the formalism of splines isobtained.

1.2.7 Conclusion on Kriging

Three different ways of calculating a Kriging interpolator have been discussed

• using the basic approach where Kriging is calculated at each location as a linearcombination of the data (Eq. 1.2)

• using Eq. 1.7, where the expression of dual Kriging, or more generally of radialbasis functions, is used

12 O. Dubrule

• minimizing Eq. 1.14 in the discrete case, where Kriging values are calculated ona discrete grid by a minimization incorporating a regularization and a distance tothe data term.

Kriging, although derived using a probabilistic formalism, is still a deterministictechnique, in the sense that one unique or “best” model is produced, In most cases,Kriging provides a representation that is very smooth. As a result the application ofnon- linear operators to Kriged models will provide biased results (Dubrule 2003).This is one of the reasons for the success of conditional simulation.

1.3 Stochastic Aspects of Geostatistics:Conditional Simulation

With conditional simulation, the approach is stochastic. A large number of realiza-tions are generated, which match the data (if the simulation is conditional) and sharethe first (mean) and second order (stationary covariance or generalized covariance)moments of the modeled random function. The main benefit of conditional simulationis that it produces realizations that behave away from the well data the same way asthe well data themselves (Dubrule 2003). This is not true with Kriging, which pro-duces a model that is smoother away from the wells than it is at the wells.

Conditional simulation can also be regarded as a technique for generatingrealizations of the conditional multigaussian pdf fully characterized by Eqs. 1.17and 1.18. In other words, the realizations “vibrate” around their Kriging mean witha variance at each location equal to the Kriging variance.

A number of conditional simulation algorithms have been developed (Chilès andDelfiner 2012, p. 478). Among them, two are routinely used in the petroleumindustry and are particularly interesting in relation with the inversion of seismic andproduction data.

1.3.1 Method 1: “Smooth Plus Rough” or“Rough Plus Smooth” Algorithm

Z xð Þ can be simply written as the sum of Kriging plus the Kriging error

Z xð Þ= Zk xð Þ+ Z xð Þ−Zk xð Þð Þ ð1:23Þ

The “smooth plus rough” (Oliver 1996) simulation method writes a conditionalsimulation Zcs xð Þ as the sum of Kriging plus a simulation of the Kriging error.A non-conditional simulation Zncs xð Þ of Z xð Þ is generated first, which honors the

1 Kriging, Splines, Conditional Simulation, Bayesian Inversion … 13

mean and the covariance of Z xð Þ, then the conditional simulation Zcs xð Þ is calcu-lated as

Zcs xð Þ= Zk xð Þ+ Zncs xð Þ−Zncsk xð Þð Þ ð1:24Þ

where Zncsk xð Þ designates Kriging of Zncs xð Þ using as data the values Zncs xið Þ of thenon-conditional simulation at the conditioning data locations. Thus to the smoothterm Zk xð Þ is added the rough term Zncs xð Þ− Zncsk xð Þð Þ. Chilès and Delfiner (2012,p. 495) show that Zcs xð Þ honors the data and has the same (generalized) covarianceas Zncs xð Þ (and hence as Z xð Þ).

Equation 1.24 can also be expressed in the form of a “rough plus smooth”equation

Zcs xð Þ= Zncs xð Þ+ Zk xð Þ−Zncsk xð Þð Þ ð1:25Þ

Using Eq. 1.17, Eq. 1.25 can be written in the discrete case, assuming that thedata are average values of the gridded values and are affected by a measurementerror. At location u of the discrete grid

zucs = zuncs +CF′ FCF′ +Cε

� �− 1y−Fzuncsð Þ ð1:26Þ

Equation 1.26 shows that conditional simulation is obtained by adding to anon-conditional simulation a Kriging of the mismatch y − Fzuncsð Þ between thedata and the unconditional simulation at the data location. This formalism willappear to be quite general and will facilitate the understanding of the relationshipbetween conditional simulation and Kalman Filtering.

1.3.2 Method 2: Sequential Gaussian Simulation (SGS)

SGS (Deutsch and Journel 1998) is probably the most popular and flexible con-ditional simulation technique used in applications. SGS works under the multi-gaussian assumption and sequentially draws random locations within the simulatedgrid. At each new random location, the value is first Kriged from the previouslysimulated values and the well data. Then, a random value is sampled from theGaussian pdf with mean equal to the Kriged value and variance equal to the Krigingvariance (SGS uses the property that, in the multivariate normal case, univariateconditional distributions are also Gaussian). Then the sampled value is merged withthe rest of the dataset, and a new random location is chosen within the simulatedgrid. The grid points where a data point is present are treated the same way as gridpoints with no data if the error ε affecting the data is different from zero. If all thedata are exact, then the grid nodes with data points are left unchanged. The result is

14 O. Dubrule

a Gaussian realization constrained by the data values and satisfying the inputstatistics (mean and covariance function).

The main difference between “rough plus smooth” and SGS is that SGS workssequentially, grid point by grid point. The sequential nature of SGS is well suited tothe geostatistical inversion of seismic data. Indeed, at each grid node, the sequentialapproach can make sure that the sampled value is compatible with both the pre-viously generated points and the seismic data at the same location, thus combiningthe advantage of single trace inversion with that of spatial coupling. This will bediscussed in Sect. 1.4.

1.3.3 Spectrum and Conditional Simulation

Since the frequency spectrum is the Fourier transform of the covariance (Chilès andDelfiner 2012, p. 66), the spectrum of a conditional simulation is the same as that ofthe data. Conditional simulation addresses the following statement from Claerbout(2002) about seismic data interpolation: Of all the assumptions we could make to fillempty bins, one that people usually find easiest to agree with is that the spectrumshould be the same in the empty-bin regions as where bins are filled.

Claerbout (2002) also defines the Prediction Error Filter (PEF) as the linearoperator T that transforms the data into a white noise. In other words, T 0T is theinverse of the covariance. Based on Eq. 1.11, this also means that T is the splineoperator associated with the covariance of the data. Claerbout (2002) shows thatunconditional simulations can be generated by applying T − 1 to a white noise. Thisis the same technique as that used by Oliver (1988) and Oliver (1995) who applieswhat he calls the square root of the covariance function to a white noise.

1.4 Geostatistical Inversion of Seismic Data

1.4.1 Deterministic Seismic Inversion

Until the mid-nineties or so, most seismic inversion studies were deterministic, inthe sense that they generated a single “best” model, usually at the same resolutionas the seismic data. Often, regularization-based or Bayesian methods were used,which led to the generation of one “maximum posterior” or “optimal for a givennorm (often L2)” 3-D acoustic impedance model (Tarantola 2005).

If the seismic inversion problem is linearized as with Fatti et al.’s (1994) model,the reflection coefficient r θð Þ at seismic time t for a seismic block of offset θ can bewritten

1 Kriging, Splines, Conditional Simulation, Bayesian Inversion … 15

r θð Þ= a1 θð Þ ∂LogIp tð Þ∂t

+ a2 θð Þ ∂LogIs tð Þ∂t

ð1:27aÞ

and y θð Þ=w θð Þ*r θð Þ+ ε θð Þ ð1:27bÞ

where Ip tð Þ and Is tð Þ are the compressive and shear impedances at time t, a1 θð Þ anda2 θð Þ are offset-related parameters, w θð Þ is the seismic wavelet for offset θ and ε θð Þis noise. This model is linear in the logarithm of Ip tð Þ and Is tð Þ. Thus, as long as thelogarithms of impedances are inverted, the seismic amplitudes can we written as inEq. 1.15 as a linear function of the logarithms of impedances, and the posteriormean obtained by multigaussian Bayesian seismic inversion (Eqs. 1.17 and 1.18) isidentical to Kriging. The solution can also be regarded as a regularization-basedsolution, where the norm controlling the smoothness is derived from the inverse ofthe covariance.

At the time when only deterministic inversion was used, geostatisticians oftentreated seismic data as “soft” information, making use only of statistical correlationsbetween seismic and reservoir parameters in order to constrain the earth models.This “soft” approach to seismic data allowed the development of some interestinginterpolation techniques such as external drift or collocated coKriging (Dubrule2003). However it also led to reservoir models not fully compatible with the seismicdata as, if a seismic forward model such as that of Eq. 1.27 was applied to them, theactual seismic data was not recovered.

The above approaches proved sufficient until the late eighties or so, as seismicdata were used at rather large scale. Thanks to the development of 3-D earthmodeling at the reservoir scale in the early nineties, it became necessary to workwith models at higher resolution than seismic data, and hence to quantify theuncertainty attached to these models. Then the availability of 4D seismic data alsocalled for new technology to better constrain the earth models. Geostatisticalinversion, described below, was developed with these issues in mind.

1.4.2 Geostatistical Inversion (GI)

The original GI algorithm (Bortoli et al. 1992; Haas and Dubrule 1994) used SGSto simulate high-resolution acoustic impedance traces constrained by seismic data.SGS starts by picking a random cell within a regular two-dimensional grid. At thiscell, a large number of possible acoustic impedance vertical traces are generated bySGS, then the trace that best matches the actual seismic trace at this location isselected. Then SGS moves to another random location of the two-dimensional grid,etc. until the whole model is filled with high-resolution impedance traces. Ini-tially SGS appeared to be well suited to this application, as it allowed the use of anykind of forward model—linear or not—relating the acoustic impedance trace gen-erated by SGS to the seismic amplitude trace at the same location. The acoustic

16 O. Dubrule

impedance vertical traces simulated by SGS typically have higher frequency con-tent than the seismic amplitudes, which makes them non-unique. This uncertaintycan be quantified by generating multiple conditional simulations. Unfortunately theuse of SGS proved to take too much computer time for large seismic datasets.

By revisiting the above GI algorithm in a Bayesian framework and in the linearcontext of Fatti’s model (Eq. 1.27), authors such as Buland and Omre (2003) orEscobar et al. (2006) not only clarified the GI formalism but also provided astraightforward conditional simulation algorithm based on Eqs. 1.17 and 1.18which was more efficient then SGS for sampling acoustic impedance traces com-patible with seismic amplitudes. Whilst Bayesian inversion provided an expressionof the posterior mean and covariance of the impedances multiGaussian pdf, GIallowed the sampling of reservoir-scale impedance realizations from this pdf.

As convincingly shown by Francis (2006a, b) or Escobar et al. (2006), cut-offoperations such as those used to translate acoustic impedance into facies can beapplied to GI realizations, thus avoiding statistical bias if these cut-offs were appliedto Kriging.

1.5 Kalman Filtering and Ensemble Kalman Filtering

1.5.1 Kalman Filtering (KF)

Suppose that a Gaussian random vector Zt− 1 has evolved until time t− 1ð Þ and thatZt− 1 is an unbiased estimate of the unknown true state vector zt− 1 at time t− 1ð Þ

Zt− 1 = zt− 1 +Rt− 1 withE Rt− 1ð Þ=0 andVar Zt− 1ð Þ=Var Rt− 1ð Þ=Ct− 1 ð1:28Þ

If the model error is neglected, the forward model relating the true state vector attime t− 1ð Þ with the state vector at time t is assumed to be a linear function Lt

z1 = Ltzt− 1 ð1:29Þ

At time step t, the unknown true state of the system has evolved according toEq. 1.29 and a vector dt of n new data may also be available. Assume that thesedata are linear functions of the state vector zt, and can be expressed as in Eq. 1.15

dt =Ftzt + εt ð1:30Þ

where the error vector εt has mean zero and covariance matrix Cεt .KF (Kalman 1960) aims to combine the information provided about zt by the

forward model Lt applied to the estimate Zt− 1 (Eq. 1.29) with the informationprovided by the data dt (Eq. 1.30). Bayes can be used for this, LtZt− 1 playing therole of the prior distribution. It is easy to verify that the covariance of the random

1 Kriging, Splines, Conditional Simulation, Bayesian Inversion … 17

vector LtZt− 1 is Ct =LtCt− 1L0t. Hence, under Gaussian assumptions the best esti-

mate is (from Eq. 1.19)

Zt = LtZt− 1 +Λt dt −FtLtZt− 1ð Þ ð1:31Þ

where the kriging weights matrix Λt (as in Eq. 1.20) is now called the Kalman gain

Λt =CtF′

t FtCtF′

t +Cεt

� �− 1 ð1:32Þ

Zt− 1 as defined in Eq. 1.28 can represent any kind of unbiased estimate based onall the information available at time t− 1ð Þ. Kriging and conditional simulation areboth unbiased estimates of zt, only their variance is different and is of courseminimum if Zt− 1 is Kriging and larger if Zt− 1 is simulation. Chilès and Delfiner(2012) (p. 497) show that the variance of the difference between a random functionand its conditional simulation is twice the Kriging variance. In case Zt− 1 is sim-ulation, Eq. 1.31 looks like the “rough plus smooth” method (Eq. 1.26) with LtZt− 1

playing the role of the non conditional simulation. Equation 1.31 makes the esti-mate LtZt− 1 conditional to the new data dt by adding an interpolation of themismatch between FtLtZt− 1 and the data.

In standard geostatistical applications, the observations are often spatial andhence assimilated simultaneously, while KF processes information sequentially,time step after time step. Tarantola (2005) (in Appendix 6.18) shows that, if in alinear least-squares problem the dataset can be divided into subsets with zerocovariance between them, then solving one global inverse problem is equivalent tosolving a series of smaller problems using the posterior state and covariance matrixof each partial problem as prior information for the next. Oliver et al. (2008) alsoshow (in Chap. 11) that, under the same assumptions as Tarantola (2005), the stepby step computation of KF provides (in the multigaussian case) the same result aswould be obtained by integrating all the data in one single step. In the case where Ltis the identity function, these two results also imply that simple Kriging wouldprovide the same result if data were incorporated sequentially into the Krigingsystem, or in one single batch (under the assumptions that each batch of data haszero covariance with the others).

1.5.2 Constraining Reservoir Models by Production Data

Fluid flow models are strongly non-linear, and linear approximations such as thosealready discussed for seismic modeling or KF cannot be used.

A distinction must be made between “history-matching”, where a single reser-voir model is modified until the flow simulation matches the production data, and“constraining reservoir models by production data”, where reservoir model

18 O. Dubrule

realizations compatible with production data are generated. Here the discussionfocuses on the second objective rather than the first one. Some techniques toaddress this objective are based on rigorous approaches such as Markov-ChainMonte Carlo (MCMC) or Genetic Algorithms (GA) (Oliver et al. 2008). But theseare very time-consuming and often unpractical. Ensemble Kalman filtering appearsto be a more practical approach for incorporating production data into the reservoirmodel.

1.5.3 Ensemble Kalman Filtering (EnKF)Versus Conditional Simulation

EnKF (Evensen 2007; Oliver et al. 2008) starts with an ensemble of initial real-izations that are not constrained by production data. Typically the state vector zt attime t contains permeabilities, porosities, saturations, pressures, and thermody-namic variables at the simulator grid nodes followed by a vector of predictedproduction data at each well i at time t.

The following notation is used for a given state vector zt

zt = z1ut, z2ut , . . . z

kut , q

1*it , . . . .q

l*it

� � ð1:33Þ

It is assumed that there are k gridded variables in zt, that the simulator grid iscomposed of p cells u, and that there are n wells i each with l new production data attime t. The total size of the state vector zt is kp+ nl. The predicted data vector d*t isthe vector of size nl

d*t = q1*1t , . . . .q1*nt , q

2*1t , . . . .q

2*nt , . . . , q

l*1t, . . . .q

l*nt

� � ð1:34Þ

The relation between state vector and predicted data is

d*t =Pzt withP= ðOnlxkp, InlxnlÞ ð1:35Þ

P is a nl× kp+ nlð Þ matrix. The function ft, which represents the flow simulator,is non-linear. If the model errors are neglected

zt = ft zt− 1ð Þ ð1:36Þ

does not modify the rock properties (unless they are affected by changes in pressureand saturation), but replaces the pressure, saturation, and simulated data with newvalues at time t.

1 Kriging, Splines, Conditional Simulation, Bayesian Inversion … 19

The problem is now to calculate the best estimate of the state vector zt com-bining the information provided by the flow simulation forward model ft zt− 1ð Þ andthat provided by the new data dt.

If ft is a linear function, this is the standard KF domain of application andEq. 1.31 applies, Lt playing the role of ft. But now ft is non-linear. It would still beconvenient to update the state vector through a generalization of Eq. 1.31

zt = ft zt− 1ð Þ+Λt dt −Pft zt− 1ð Þð Þ ð1:37Þ

where the Kalman gain Λt is obtained using Eq. 1.32. Assuming that there is noerror associated with the data, Eq. 1.32 can be simplified into

Λt =CtP′ PCtP′� �− 1 ð1:38Þ

Equation 1.38 requires the knowledge of the covariance Ct of ft zt− 1ð Þ, in otherwords the covariance of the image of the state vector after application of the flowsimulation model ft. ft is non-linear and this covariance cannot be simply calculated—as in the linear case—from the covariance at the previous step. EnKF addressesthis issue by statistically deriving this covariance using the information from themultiple realizations, typically about a hundred of them. This is the key idea behindEnKF.

There are of course a number of issues resulting from the fact that the covari-ances are calculated from a finite number of realizations of the ensemble. The firstone is spurious correlation, because the ensemble members are not independentexcept in the starting ensemble. The second one is that if the number of realizationsin the ensemble is not large enough, then the covariances are poorly estimated.Standard geostatistics addresses this by fitting mathematical models to the experi-mental covariances, in order to smooth the spurious correlations.

1.5.4 Ensemble Kalman Filtering and Its Relationshipwith CoKriging

In Eq. 1.37, focus now on the rock properties in the state vector. ft zt− 1ð Þ leaves therock properties unchanged, as only the time-dependent state vectors in the simulatorgrid are calculated by one time-step of the flow simulator, whilst Λt dt −Pft zt− 1ð Þð Þis a linear combination of the differences between observed and predicted pro-duction data at each well. Thus EnKF interpolates between the wells by calculatinga linear combination of these differences across the field, then adds these interpo-lated difference to the rock properties model. Is it possible to reformulate EnKF as awell by well geostatistical approach?

The term Λt in Eq. 1.37 is the Kalman gain as given by Eq. 1.38. In the casewhere there is no error affecting the data, Eqs. 1.37 and 1.38 can be written

20 O. Dubrule

zt − ft zt− 1ð Þ=CtP′ PCtP′� �− 1

dt −Pft zt− 1ð Þð Þ ð1:39Þ

The left-hand side is the update calculated by EnKF for the property of interestas the time step evolves from t− 1ð Þ to t. The Kalman gain coefficients of theright-hand side are nothing else than the simple coKriging weights (see for instanceChilès and Delfiner 2012, p. 303).

Thus, each estimate of a 3-D spatial parameter such as porosity or permeabilityat time t− 1ð Þ is updated at time t by a linear combination of all the inconsistenciesgenerated by this parameter at the data points. Since, in the case of flow simulations,many parameters are involved in the production profiles prediction, all the indi-vidual parameters’ 3-D models must be corrected in a consistent way, which is whymultivariate coKriging—and not univariate Kriging applies here.

1.6 Beyond the Formal Relationship Between Geostatisticsand Bayes

1.6.1 Two Identical Formalisms but Different Assumptions

The above developments show that techniques such as conditional simulation,Bayesian inversion, geostatistical inversion and ensemble Kalman filtering follow asimilar mathematical formalism.

However, their philosophy of application differs in the way the covariance isapproached. This can be understood by looking again as Bayes rule as presented inEq. 1.16

fpost zð Þ∝fprio zð Þg y ̸zð Þ ð1:40Þ

With geostatistics, the experimental (generalized) covariance calculated on thedata y is fitted by a model which becomes the covariance of the unconditionaldistribution fprio zð Þ. Then the data y are used a second time through the simulationconditioning process of Eq. 1.26.

With Bayes, the covariance model associated with fprio zð Þ is a prior based onlocal or analog knowledge, but not on the data themselves (Tarantola 2005). Thisprior is transformed into a posterior covariance through the conditioning process ofEq. 1.40.

With geostatistics, the aim of conditional simulation is to generate realizationsthat match the data and satisfy the input covariance; the SGS and rough plus smoothalgorithms work only if the data themselves satisfy this input covariance. But therandom function Zcs xð Þ of Eqs. 1.24 and 1.25 is not an ergodic or even a stationaryrandom function; its variance at each location x is equal to the Kriging variance andchanges with x, as it is zero at the data points. In other words, the covariance of therandom function Z xð Þ is different from that of Zcs xð Þ conditionally to the data

1 Kriging, Splines, Conditional Simulation, Bayesian Inversion … 21

(Chilès and Delfiner 2012, p. 497). But the covariance calculated on a singleconditional realization does not “see” any difference between the grid cells asso-ciated with data points and those not associated with data points. It is only as therealizations change, leaving the data unchanged, that the covariance across real-izations appears non-stationary and hence non-ergodic.

On the other hand, Bayes combines a prior covariance—usually different fromthat of the data—with a data-based likelihood, resulting into a posterior pdf that sitssomewhere between the prior and the likelihood. Bayes updates prior covariancesbased on new data whilst conditional simulation anchors the realizations against thehard data (Escobar, personal communication).

1.6.2 Model Falsifiability

Tarantola (2006) challenges the geostatistical and Bayes formalisms if models areto be falsifiable or have a scientific meaning: I suggest that the setting, in principle,for an inverse problem should be as follows: use all available prior information tosequentially create models of the system, potentially an infinite number of them. Foreach model, solve the forward modeling problem, compare the predictions to theactual observations and use some criterion to decide if the fit is acceptable orunacceptable, given the uncertainties in the observations and, perhaps, in thephysical theory being used. The unacceptable models have been falsified, and mustbe dropped. The collection of all the models that have not been falsified representthe solution of the inverse problem. Thus, Tarantola (2006) offers to keep all theprior realizations that are compatible with the data. Thus the data are used tovalidate or reject the prior realizations, rather than update the prior pdf into theposterior.

1.6.3 Looking Ahead: Machine Learning and Falsifiability

The fast growth in machine learning algorithms (Goodfellow et al. 2016) is chal-lenging the geostatistical and Bayesian formalisms in situations where data areplenty. Thanks to this large number of data, the approach used to falsify a con-volutional neural network model (for instance) relating input parameters to data isoften to test whether the convolutional model works as well on a training (orcalibration) dataset as on a test dataset not used for training. The prior model itselfis completely data-driven, which contradicts Tarantola (2006) but the validationstep is along the lines of his above recommendations! This topic is likely to gen-erate interesting discussions in the future.

22 O. Dubrule

1.7 Conclusion

The objective of this chapter was to discuss the convergence observed over the lastfifty years between geostatistics and other modelling and inversion techniques.

A formal convergence exists between the main techniques used to constrainreservoir models by multi-disciplinary data. Kriging, splines, conditional simula-tion, geostatistical inversion and ensemble Kalman filtering can be interpreted usingeither the geostatistical formalism or Bayes.

Most of these techniques amount to the same approach where an initial model isupdated by using a linear combination of the mismatches between the new data andtheir prediction from the initial model (Eqs. 1.19, 1.26, 1.31 and 1.39).

However the methods above have a different philosophy towards the inferenceof the covariances used in these calculations. Bayes uses the data to update a priorpdf which is independent of the data. Geostatistics generate realizations of condi-tional simulations that reproduce the modeled covariance—or the spectrum—of thedata. EnKF does not model a covariance but directly uses the empirical covariancesderived from the ensemble realizations and their flow simulations.

Acknowledgements The author would like to thank Imperial College London, and Total forseconding him there as a Visiting Professor. Igor Escobar and Danila Kuznetsov are thanked forthe fruitful discussions held at the Total Geoscience Research Centre in Aberdeen (UK). Theauthor will also never forget the passionate and illuminating discussions with Albert Tarantola atthe Café Beaubourg in Paris!

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24 O. Dubrule